Proof Regarding a Twice Differentiable Function

Context: We are learning Rolle, Lagrange, Fermat, Taylor Theorems in our Real Analysis class. We just finished continuity and are now studying differentiation.

Question:
Let f: [a,b] --> R, a < b, twice differentiable with the second derivative continuous such that f(a)=f(b)=0.

Denote M = sup |f "(x)| where x is in [a,b]

and g:[a,b] --> R, g(x)=(1/2)(x-a)(b-x)

i) Prove that for all x in [a,b], there exists

Cx in (a,b) such that f(x)= - f "(Cx)g(x).

ii) Prove that if there exists x0 in (a,b) such that

|f(x0)| = Mg(x0), then f = Mg or f=-Mg.

Notes:
Cx is a constant dependent on x ,

x0 is a particular x in (a,b)

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...re is the detailed proof.
Note , . So we have , and for all .
Now consider the function . We have
,
We also note . This implies that is decreasing in the interval . Now I claim for all . If not, we have some with . Since , by the Mean Value Theorem, there exists a , such that . On the other hand, , so there exists a , such that . But and , this contradicts ...